Part II Long–Range Percolation
4.2 Brownian paths homogeneously distributed in space
The model
For λ > 0, let (Ω,A,Pλ) be a probability space on which a Poisson point process E
with intensityλ×Lebd is defined, where Lebd is the d-dimensional Lebesgue measure.
Conditionally onE, we fix a collection of independent Brownian motions{(Bx
t)t≥0, x∈
E}such thatBx
0 =xfor each x∈ E and such that (Btx−x)t≥0 is independent ofE. For
t, r≥0 we study theoccupied set
Ot,r:= [ x∈E [ 0≤s≤t B(Bsx, r), (4.2.1)
where B(y, r) denotes the ball with respect to the euclidean norm around y ∈Rd with radius r. If d≤ 3, then we put r = 0. The reason for that will become clearer when discussing the results. In the remainder of this section we writeOtinstead ofOt,0.
We are interested in the percolative properties ofOt,r: Is there an unbounded cluster
for larget? Is it unique? What happens for smallt? Since an elementary monotonicity argument shows thatt7→ Ot,r is non-decreasing, the first and the third question may be
rephrased as follows: Is there a percolation transition int?
Motivation and related models
The model described above fits into the class of continuum percolation models, which have been studied intensively by both mathematicians and physicists. Their first appearance can be traced back (at least) to Gilbert [G61] under the name of random plane networks. Gilbert was interested in modeling infinite communication networks of stations with rangeR > 0. He did this by connecting each two points of a Poisson point process on R2 whenever their distance is less than R. Another application, mentioned in his work is the modeling of a contagious infection. Here, each individual gets infected when it has distance less thanR to an infected individual.
A subclass of continuum percolation models follows the following recipe: Consider a point process (e.g. a Poisson point process) and attach to each of its points a geometric object, like a disk of random radius (Boolean model) or a segment of random length and random orientation (Poisson sticks model or needle percolation). Our model also falls into
4.2 Brownian paths homogeneously distributed in space
this class: to each point of a Poisson point process we attach a Brownian path (a path of a Wiener sausage whend≥4). This can be seen as a model of defects that are randomly distributed in a material and are propagating at random. We can think for example of an (infinite) piece of wood containing (homogeneously distributed) worms, where each worm eats its way through the piece of wood at random (see Menshikov, Molchanov and Sidorenko [MMS88] for other physical motivations of continuum percolation). The informal description above is reminiscent of (and actually borrowed from) the problem of the disconnection of a cylinder by a random walk, which itself is linked to interlacement percolation [S10]. The latter is defined as the random subset obtained when looking at the trace of a simple random walk on the torus (Z/NZ)d, starting from the uniform
distribution and running up to time uNd in the limit as N ↑ ∞. Here, uplays the role
of an intensity parameter for the interlacement set. However, even though the model of random interlacements and our model seem to share some similarities, there is an important difference: in the interlacement model the number of trajectories that enter a ball of radiusR scales like cRd−2 for somec >0, whereas in our model it is at least of orderRd.
Another motivation for studying our model is that it should arise as the scaling limit of a class of discrete dependent percolation models, namely a system of independent finite-time random walks homogeneously distributed onZd. The latter can also be seen as a system of non-interacting ideal polymer chains.
Results
Fixλ >0.
Theorem 4.2.1. [No percolation for d= 1] Let d= 1. Then, for all t≥0, the set
Othas almost surely no unbounded cluster.
Theorem 4.2.2. [Percolation phase transition and uniqueness for d= 2,3] Let d= 2,3. Then there exists atc=tc(λ, d)>0 such that, fort < tc,Othas almost surely
no unbounded cluster whereas, for t > tc, Ot has almost surely a unique unbounded
cluster.
Letd≥4 andr >0. We denote by λc(r) the critical value such that for all λ < λc(r)
the setO0,r almost surely does not contain an unbounded cluster, whereas forλ > λc(r)
it does. Gou´er´e [G08] showed thatλc(r)>0 forr >0 and limr→0λc(r) =∞.
Theorem 4.2.3. [Percolation phase transition and uniqueness for d ≥ 4] Let d≥ 4, and let r >0 be such that λ < λc(δr). Then there exists a tc = tc(λ, d, r) > 0
such that, for t < tc, Ot,r has almost surely no unbounded cluster whereas, for t > tc,
Ot,r has almost surely a unique unbounded cluster.
Comments on the results
Theorems 4.2.1–4.2.3 describe a phase transition in t. It would be possible to play with the intensity λ instead. Indeed, when we multiply the intensity λ by a factor η
we change the typical distance between two Poisson points by a factor η−1/d. By scale
invariance of Brownian motion, the percolative behaviour of the model is the same when we consider the Brownian paths up to timeη−2/dt instead. Hence, tuningλboils down
to tuningt.
It is worthwhile to mention that Theorem 4.2.2 is stated only in the caser= 0, which is the case of interest to us. The result is the same whenr >0, up to minor modifications. However, ifd≥4, then the paths of two independent d-dimensional Brownian motions starting at different points do not intersect andrhas to be chosen positive, otherwise no percolation phase transition occurs.
To sum up, the above settle the first questions typically asked when studying a new percolation model. Many challenges are open. One may wonder, for instance, how fast is the decay of the probability (in the supercritical regime) that a ball of a certain size, centered at the origin, is contained in the vacant set. Moreover, it would be interesting to investigate the scaling behaviour oftc in dimensiond≥4 asrtends to zero. One could
ask for sharp upper and lower bounds ontc. Finally, it is not clear whether percolation
occurs attc or not.